Schrodinger wave equation: Time Dependent and Time Independent Wave Equation


What is the Schrodinger wave equation? Derive Schrodinger`s time dependent and time independent wave equation.

Write Schrodinger`s time dependent and time independent wave equation. Explain its physical significance and discuss the term in equation which is related with physical problem.

Answer:

In the year 1926 the Austrian physicist Erwin Schrödinger describes how the quantum state of a physical system changes with time in terms of partial differential equation. This equation is known as the Schrodinger wave equation. In quantum mechanics, the analogue of Newton’s law of motion is Schrodinger equation for a quantum system (usually atoms, molecules, and subatomic particles whether free, bound, or localized). It is not a simple algebraic equation, but in general a linear partial differential equation, describing the time-evolution of the system’s wave function. So the Schrodinger wave equation is a equation, which expresses in the form of wave function ψ of matter waves in different physical conditions.

Time dependent Schrodinger’s wave equation:

Consider a particle of mass m moving in positive x direction. The potential energy of the particle is V, momentum is p and total energy is E. So the free particle wave equation is:

Schrodinger wave equation

Schrodinger time dependent wave equation:

The total energy is the sum of kinetic energy and potential energy; so the total energy of the particle is

Schrodinger time dependent wave equation

This is Schrodinger’s time dependent wave equation in one dimension form and In three-dimension.

The Schrodinger equation is the name of the basic non-relativistic wave equation used in one version of quantum mechanics to describe the behavior of a particle in a field of force. There is the time dependent equation used for describing progressive waves, applicable to the motion of free particles.

Time independent Schrodinger’s equation:

The Schrodinger’s time independent wave equation describes the standing waves. Sometimes the potential energy of the particle does not depend upon time, and the potential energy is only the function of position. In such cases, the behavior of the particle is expressed in terms of the Schrodinger’s time independent wave equation.

According to classical mechanics, the total energy of the particle is

 Schrodinger time independent wave equation

This is time independent Schrodinger equation for one-dimension motion of particle. For three-dimensional motion

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